Open loop TF and ramp error from (Ks+b)/(s²+as+b)

Open-loop transfer function

For unity feedback, $T(s) = \dfrac{G(s)}{1 + G(s)}$, so

$$G(s) = \frac{T(s)}{1 - T(s)}$$ $$1 - T(s) = \frac{s^2 + as + b - Ks - b}{s^2 + as + b} = \frac{s^2 + (a - K)s}{s^2 + as + b}$$ $$G(s) = \frac{Ks + b}{s^2 + (a - K)s} = \frac{Ks + b}{s\left[s + (a - K)\right]}$$

The open-loop system is type 1 (one pole at the origin).

Steady-state error for unit ramp

For a unit ramp $R(s) = 1/s^2$, $e_{ss} = \dfrac{1}{K_v}$ where the velocity error constant is

$$K_v = \lim_{s\to 0} sG(s) = \lim_{s\to 0}\frac{Ks + b}{s + (a - K)} = \frac{b}{a - K}$$

Therefore

$$e_{ss} = \frac{1}{K_v} = \frac{a - K}{b}$$

Direct check from the error transfer function

$$E(s) = R(s)\left[1 - T(s)\right] = \frac{1}{s^2}\cdot\frac{s\left[s + (a - K)\right]}{s^2 + as + b}$$ $$e_{ss} = \lim_{s\to 0}sE(s) = \lim_{s\to 0}\frac{s + (a - K)}{s^2 + as + b} = \frac{a - K}{b}$$

Hence proved (valid when the closed loop is stable, i.e. $a > 0$, $b > 0$). Choosing $K = a$ makes the ramp error zero, because the open loop then becomes type 2.